Let E be an elliptic curve over $\mathbb{F}_p$. Suppose that $t^2-4p\not\equiv 0\pmod{l}$, where $l$ is an Elkies prime and $t$ the norm of Frobenius endomorphism $\varphi$. Then $\varphi$ acts on $E[l]$ as a matrix $\left( \begin{smallmatrix} \alpha &0\\ 0&\beta \end{smallmatrix} \right)$
and let $T\subset PGL_2(F_l)$be a maximal torus containing $\varphi$. In other words $T=\{\left( \begin{smallmatrix} \alpha &0\\ 0&\beta \end{smallmatrix} \right) | \alpha, \beta \in F_l^{*}\}$. Then the image $\bar{T}$ of $T$ in $PGL_2(F_l)$ is cyclic of order $l-1$ and isomorphic to $F_l^{*}$
When $T$ is non-split, it is isomorphic to $F_{l^2}^{*}$ and the order of $\bar{T}$ is $l+1$.
I can't see how $\bar{T}$ is cyclic of order $l+1$.
Schoof, (1995) Counting points on eliptic curve over finite field page no:240.proposition 6.3
Please help me to understand this.